We report on a study of cold reactive collisions between sympathetically cooled molecular ions and laser-cooled atoms in an ion-atom hybrid trap. Chemical reactions were studied at average collision energies <E(coll)>/k(B)>/~20 mK, about 2 orders of magnitude lower than has been achieved in previous experiments with molecular ions. Choosing N(2)(+)+Rb as a prototypical system, we find that the reaction rate is independent of the collision energy within the range studied, but strongly dependent on the internal state of Rb. Highly efficient charge exchange four times faster than the Langevin rate was observed with Rb in the excited (5p) (2)P(3/2) state. This observation is rationalized by a capture process dominated by the charge-quadrupole interaction and a near resonance between the entrance and exit channels of the system. Our results provide a test of classical models for reactions of molecular ions at the lowest energies reached thus far.